Observation Nuclei in Affine Geometry: Symbolic Powers, Fat Points, and Infinitesimal Recovery

Abstract

This paper integrates point-dependent infinitesimal data into affine geometry through quantale nuclei and finitary weak ideal systems. For a subset $X \subseteq k^n$ and a bounded function $\nu:X \to \mathbb{N}_{>0}$, we study the observation nucleus $\Phi^*_{X,\nu}(A) = \bigcap_{a\in X} (A+\mathfrak{m}_a^{\nu(a)})$. This construction organizes several classical interfaces within one point-dependent operator: closed observations recover symbolic powers through the Zariski-Nagata theorem, and finite observations restrict on ideals to translation by a fat-point ideal through the Chinese remainder theorem. The principal uniform identity is the exact relative Nullstellensatz $\sqrt{\Phi^*_{X,\nu}(I)} = \mathcal{I}(X\cap V_{\mathrm{aff}}(I))$, which separates the reduced geometry determined by the observation set from the retained infinitesimal multiplicities determined by $\nu$. This identity yields a density criterion for fixed prime ideals and identifies their space with the sobrification of the observed point set. For constant full-affine observation order, the descending tower has a common radical $\sqrt{I}$, and the nilpotent Nullstellensatz of Eisenbud and Hochster implies finite-stage stabilization: for every ideal $I$, there exists $e \geq 1$ such that $\Phi^*_q(I) = I$ for all $q \geq e$. The observation formalism therefore gives a simultaneous closed-point congruence interpretation of this classical theorem. Finally, the full affine nucleus is not a fixed-ideal translation in any positive dimension, and an explicit unbounded example shows that boundedness cannot be omitted from the general radical formula.